Best Known (58, 73, s)-Nets in Base 4
(58, 73, 1048)-Net over F4 — Constructive and digital
Digital (58, 73, 1048)-net over F4, using
- 42 times duplication [i] based on digital (56, 71, 1048)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 20)-net over F4, using
- digital (45, 60, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 15, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 15, 257)-net over F256, using
- (u, u+v)-construction [i] based on
(58, 73, 4071)-Net over F4 — Digital
Digital (58, 73, 4071)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(473, 4071, F4, 15) (dual of [4071, 3998, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(473, 4121, F4, 15) (dual of [4121, 4048, 16]-code), using
- construction XX applied to Ce(14) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- linear OA(467, 4096, F4, 15) (dual of [4096, 4029, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(449, 4096, F4, 11) (dual of [4096, 4047, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(443, 4096, F4, 10) (dual of [4096, 4053, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(45, 24, F4, 3) (dual of [24, 19, 4]-code or 24-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(14) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(473, 4121, F4, 15) (dual of [4121, 4048, 16]-code), using
(58, 73, 1755549)-Net in Base 4 — Upper bound on s
There is no (58, 73, 1755550)-net in base 4, because
- 1 times m-reduction [i] would yield (58, 72, 1755550)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 22 300772 349800 622371 538147 425888 858348 883556 > 472 [i]